A Low-Frequency Approximation to the Maxwell Equations Simultaneously Considering Inductive and Capacitive Phenomena

Robust full‐wave Maxwell solver in time‐domain using magnetic vector potential with edge elements

“…To address the distributed parasitic effects in electromagnetic (EM) devices, such as coupled inductive and capacitive effects in the complicated solid conductor windings of high-frequency transformers and electric machines [1,2], and the fast transient voltage surges in traditional transformers due to lightening [3], it is indispensable to incorporate the displacement current in the numerical modelling. Due to transient voltage excitation with nonsinusoidal waveform, it is natural to formulate the problem in time domain.…”

“…For low-frequencies, there are also wide applications with resistive, inductive and capacitive effects [4][5][6], it also requires the study of low-frequency full-wave Maxwell solvers. Since traditional high-frequency full-wave solvers, where the electric field intensity are usually taken as the unknown variable, are not convenient to impose voltage excitations [2], the magnetic vector potential (MVP)-based formulation is good to use because the associated electric scalar potential (ESP) is also explicitly present [7]. As traditional low-frequency eddy-current solvers do not take into account displacement current, it is highly desirable to develop a time-domain full-wave Maxwell solver for analysing the abovementioned problems with both inductive and capacitive effects.…”

“…Existing MVP formulations considering both inductive and capacitive effects are either ungauged or with too complicated stabilisation procedure or with non-symmetric system matrix. For the ungauged formulation first studied in frequency domain [2] and then in time domain [12,13], due to dropping of the ω 2 εA(or ε∂ 2 A/∂t 2 ) term, the resultant system matrix is no longer symmetric. What's more, preconditioned iterative solvers have to be used to find a solution because the coefficient matrix is singular.…”

“…By the introduction of the MVP, (3) can be automatically satisfied. From the Maxwell's second (2), one can also introduce an electric scalar potential (ESP) φ such that…”

“…The formulation studied in [3][4] only use the first (10) together with the Coulomb gauge, the continuity equation is not explicitly solved. As remarked by Koch et al [2], even though the continuity equation of the total current is implicitly included in the Ampere's law (1), it is necessary to explicitly enforce it, in order to obtain a well-posed problem in terms of the potentials. Especially, at low frequencies, it is not sufficient by imposing the continuity equation in an implicit manner.…”